1. CS552: Computer Graphics Lecture 18: Representing Cubic Splines

2. Recap Parametric cubic spline Spline of order n Piecewise polynomial Cubic spline o One segment: end points and its tangent o Two segments: end points of each segment, tangent at the terminal points o K segments

3. Objective After completing this lecture students will be able to Generalize cubic spline with K segments Solve numerical problems

4. Parametric cubic spline segment Assume, n=3 Constant curvature at the internal joint between the two spans.

5. Parametric cubic spline segment Target: Calculate the tangent vector at the junction point

6. Piecewise spline

7. Basis functions To generate the cubic spline curve The magnitude of the tangent vectors is changed, the slope of the cubic segments between data points is changed. On the other hand, the direction of the tangent vectors controls the shape of the cubic segments at their points.

8. Normalized Parameters

9. Coefficient matrix

10. Normalized Parameters Dimensions of these matrices?

11. Normalized Parameters Non-zero terms in the M matrix are at the indices How to make M a square matrix?

12. Choice of boundary condition The choice of boundary condition depends upon o if only a few data points are known o if physical constraints require accurate control of the curve shape at the ends. Specify the two end tangent vectors

13. Boundary condition

14. End conditions End conditionM matrix non-zero elements B(K,1); B(K,N) ClampedM(1,1)=1; M(N,N)=1B(K,1) = U(K,1) B(K,N) = U(K,N) RelaxedM(1,1) = 1; M(N, N-1)= 2 M(1,2) = 0.5; M(N,N) = 4

15. Numerical Problem Assume that the three position vectors

16. Thank you Next Lecture: Bezier Curve